Exact Asymptotic Estimation of Unknown Parameters of Perturbed LRE with Application to State Observation

TL;DR

The paper tackles unknown parameter estimation in a linear regression with additive perturbations by introducing an averaging-based law that, when the disturbance satisfies C1-C2 averaging conditions, ensures asymptotic convergence of the parameter estimation error. Building on dynamic regressor extension and mixing (DREM), and employing both Kreisselmeier and FE-based extensions, the authors derive conditions under which the error remains bounded and converges to zero, even without strict PE. The approach is applied to state observation for nonlinear systems, where an observer is formed using the estimated parameters and a Lyapunov-based bound guarantees controlled state estimation error under disturbances. These results extend adaptive identification and state estimation capabilities to systems with averaging-type perturbations, with practical implications for robust control and observer design.

Abstract

Most identification laws of unknown parameters of linear regression equations (LRE) ensure only boundedness of a parametric error in the presence of additive perturbations, which is almost always unacceptable for practical scenarios. In this paper, a new identification law is proposed to overcome this drawback and guarantee asymptotic convergence of the unknown parameters estimation error to zero in case the mentioned additive perturbation meets special averaging conditions. Such law is successfully applied to state reconstruction problem. Theoretical results are illustrated by numerical simulations.

Figures (6)

• Figure 1: Behavior of $\Delta \left( t \right)$, ${{\@fontswitch\mathcal{W}}_i}\left( t \right)$ and comparison of $\gamma {\Delta ^3}\left( t \right) + + \Delta \left( t \right)\dot \Delta \left( t \right)\hat{\kappa} \left( t \right) + \dot \Delta \left( t \right)$ and $\eta \Delta \left( t \right)$ for $\eta = 50$.
• Figure 2: Behavior of $\hat{\theta} \left( t \right)$ for \ref{['eq6']} and \ref{['eq9']}
• Figure 3: Behavior of $\Delta \left( t \right)$, ${{\@fontswitch\mathcal{W}}_i}\left( t \right)$ and comparison of $\gamma {\Delta ^3}\left( t \right) + + \Delta \left( t \right)\dot \Delta \left( t \right)\hat{\kappa} \left( t \right) + \dot \Delta \left( t \right)$ and $\eta \Delta \left( t \right)$ for $\eta = 10$.
• Figure 4: Behavior of $\hat{\theta} \left( t \right)$ for \ref{['eq13']} + \ref{['eq6']} and \ref{['eq13']} + \ref{['eq9']}.
• Figure 5: Behavior of $\Delta \left( t \right)$, ${{\@fontswitch\mathcal{W}}_i}\left( t \right)$ and comparison of $\gamma {\Delta ^3}\left( t \right) + + \Delta \left( t \right)\dot \Delta \left( t \right)\hat{\kappa} \left( t \right) + \dot \Delta \left( t \right)$ and $\eta \Delta \left( t \right)$ for $\eta = 100$.